$($a$)(15$ points$)$ Consider the function $f(x_1,x_2)=c-(x_1-a{)}^2-(x_2-b{)}^2.$

where a$,$b$,$c are digits from your student id$.$ Run one cycle of a genetic algorithm to

maximize this function. Show and explain how and what do you compute at each step of

the genetic algorithm:

$*$Assume that the features are real$-$valued.

$*$You can choose the simplest fitness$/$selection criteria to choose two points. You can choose

simple crossover $($e$.$g$.$ average$)$ and mutation $($random noise $)$ methods as well.

$*$Start by choosing four random points in the domain of the function. E$.$g$.[0,0],[a,0],$dots,

$?,?($Do not choose $a,b).$ Be careful to bracket the maximum in a rectangle.

Initial population:

Fitness function:

Selection:

Crossover:

Mutation:

New population:

$($b$)(5$ points$)$ If you did not bracket the optimal point $($meaning your initial points did not

enclose the optimal point$),$ your run might not converge to the global maximum. Com$-$

ment on which of our simplifications might have contributed to this outcome and discuss

strategies to avoid local minima.